Theorem ntrclscls00 41206

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclscls00	⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑗,𝐾,𝑘   𝜑,𝑖,𝑗,𝑘

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclscls00

Step	Hyp	Ref	Expression
1		ntrcls.o	. . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
2		ntrcls.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝐵)
3		ntrcls.r	. . . . . 6 ⊢ (𝜑 → 𝐼𝐷𝐾)
4	1, 2, 3	ntrclsfv1 41195	. . . . 5 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
5	4	fveq1d 6670	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐾‘∅))
6	2, 3	ntrclsbex 41174	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
7	1, 2, 3	ntrclsiex 41193	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8		eqid 2738	. . . . 5 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
9		0elpw 5219	. . . . . 6 ⊢ ∅ ∈ 𝒫 𝐵
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → ∅ ∈ 𝒫 𝐵)
11		eqid 2738	. . . . 5 ⊢ ((𝐷‘𝐼)‘∅) = ((𝐷‘𝐼)‘∅)
12	1, 2, 6, 7, 8, 10, 11	dssmapfv3d 41157	. . . 4 ⊢ (𝜑 → ((𝐷‘𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
13	5, 12	eqtr3d 2775	. . 3 ⊢ (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14		dif0 4259	. . . . . . 7 ⊢ (𝐵 ∖ ∅) = 𝐵
15	14	fveq2i 6671	. . . . . 6 ⊢ (𝐼‘(𝐵 ∖ ∅)) = (𝐼‘𝐵)
16		id 22	. . . . . 6 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘𝐵) = 𝐵)
17	15, 16	syl5eq 2785	. . . . 5 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
18	17	difeq2d 4011	. . . 4 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵 ∖ 𝐵))
19		difid 4257	. . . 4 ⊢ (𝐵 ∖ 𝐵) = ∅
20	18, 19	eqtrdi 2789	. . 3 ⊢ ((𝐼‘𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
21	13, 20	sylan9eq 2793	. 2 ⊢ ((𝜑 ∧ (𝐼‘𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22		pwidg 4507	. . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
23	6, 22	syl 17	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝐵)
24	1, 2, 3, 23	ntrclsfv 41199	. . 3 ⊢ (𝜑 → (𝐼‘𝐵) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))))
25	19	fveq2i 6671	. . . . . 6 ⊢ (𝐾‘(𝐵 ∖ 𝐵)) = (𝐾‘∅)
26		id 22	. . . . . 6 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
27	25, 26	syl5eq 2785	. . . . 5 ⊢ ((𝐾‘∅) = ∅ → (𝐾‘(𝐵 ∖ 𝐵)) = ∅)
28	27	difeq2d 4011	. . . 4 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = (𝐵 ∖ ∅))
29	28, 14	eqtrdi 2789	. . 3 ⊢ ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝐵))) = 𝐵)
30	24, 29	sylan9eq 2793	. 2 ⊢ ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼‘𝐵) = 𝐵)
31	21, 30	impbida 801	1 ⊢ (𝜑 → ((𝐼‘𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1542   ∈ wcel 2113  Vcvv 3397   ∖ cdif 3838  ∅c0 4209  𝒫 cpw 4485   class class class wbr 5027   ↦ cmpt 5107  ‘cfv 6333  (class class class)co 7164   ↑_m cmap 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-map 8432

This theorem is referenced by: (None)